Results 141 to 150 of about 349,803 (277)
Sensitivity and Hamming Graphs
Journal of Graph Theory, EarlyView.ABSTRACT For any m ≥ 3 $m\ge 3$ we show that the Hamming graph H ( n , m ) $H(n,m)$ admits an imbalanced partition into m $m$ sets, each inducing a subgraph of low maximum degree. This improves previous results by Tandya and by Potechin and Tsang, and disproves the Strong m $m$‐ary Sensitivity Conjecture of Asensio, García‐Marco, and Knauer.
Sara Asensio +3 more
wiley +1 more source
Tight Bounds for Hypercube Minor‐Universality
Journal of Graph Theory, EarlyView.ABSTRACT A graph G $G$ is m $m$‐minor‐universal if every graph H $H$ with at most m $m$ edges and no isolated vertices is contained as a minor in G $G$. Recently, Benjamini, Kalifa and Tzalik proved that there is an absolute constant c>0 $c\gt 0$ such that the d $d$‐dimensional hypercube Qd ${Q}_{d}$ is (c⋅2d/d $c\cdot {2}^{d}/d$)‐minor‐universal ...
Emma Hogan +5 more
wiley +1 more source
Treewidth Versus Clique Number. V. Further Connections With Tree‐Independence Number
Journal of Graph Theory, EarlyView.ABSTRACT We continue the study of ( tw , ω ) $({\mathsf{tw}},\omega )$‐bounded graph classes, that is, hereditary graph classes in which large treewidth is witnessed by the presence of a large clique, and the relation of this property to boundedness of the tree‐independence number, a graph parameter introduced independently by Yolov in 2018 and by ...
Claire Hilaire +2 more
wiley +1 more source
An Improved Quasi‐Isometry Between Graphs of Bounded Cliquewidth and Graphs of Bounded Treewidth
Journal of Graph Theory, EarlyView.ABSTRACT Cliquewidth is a dense analogue of treewidth. It can be deduced from recent results by Hickingbotham [arXiv:2501.10840] and Nguyen, Scott, and Seymour [arXiv:2501.09839] that graphs of bounded cliquewidth are quasi‐isometric to graphs of bounded treewidth.
Marc Distel
wiley +1 more source
Upper Bounds on the Minimum Size of Feedback Arc Set of Directed Multigraphs With Bounded Degree
Journal of Graph Theory, EarlyView.ABSTRACT An oriented multigraph is a directed multigraph without directed 2‐cycles. Let fas ( D ) $\text{fas}(D)$ denote the minimum size of a feedback arc set in an oriented multigraph D $D$. In several papers, upper bounds for fas ( D ) $\text{fas}(D)$ were obtained for oriented multigraphs D $D$ with maximum degree upper‐bounded by a constant ...
Gregory Gutin +3 more
wiley +1 more source
Long Induced Paths in K s , s ${K}_{s,s}$‐Free Graphs
Journal of Graph Theory, EarlyView.ABSTRACT More than 40 years ago, Galvin, Rival, and Sands showed that every K s , s ${K}_{s,s}$‐free graph containing an n $n$‐vertex path must contain an induced path of length f ( n ) $f(n)$, where f ( n ) → ∞ $f(n)\to \infty $ as n → ∞ $n\to \infty $. Recently, it was shown by Duron, Esperet, and Raymond that one can take f ( n ) = ( log log n ) 1 /
Zach Hunter +3 more
wiley +1 more source
On Oriented Colourings of Graphs on Surfaces
Journal of Graph Theory, EarlyView.ABSTRACT For an oriented graph G $G$, the least number of colours required to oriented colour G $G$ is called the oriented chromatic number of G $G$ and denoted χ o ( G ) ${\chi }_{o}(G)$. For a non‐negative integer g $g$ let χ o ( g ) ${\chi }_{o}(g)$ be the least integer such that χ o ( G ) ≤ χ o ( g ) ${\chi }_{o}(G)\le \unicode{x0200A}{\chi }_{o}(g)
Alexander Clow
wiley +1 more source